We report that high-quality single-layer graphene (SLG) has been successfully synthesized directly on various dielectric substrates including amorphous SiO<sub>2</sub>/Si by a Cu-vapor-assisted chemical vapor deposition (CVD) process. The Cu vapors produced by the sublimation of Cu foil that is suspended above target substrates without physical contact catalyze the pyrolysis of methane gas and assist nucleation of graphene on the substrates. Raman spectra and mapping images reveal that the graphene formed on a SiO<sub>2</sub>/Si substrate is almost defect-free and homogeneous single layer. The overall quality of graphene grown by Cu-vapor-assisted CVD is comparable to that of the graphene grown by regular metal-catalyzed CVD on a Cu foil. While Cu vapor induces the nucleation and growth of SLG on an amorphous substrate, the resulting SLG is confirmed to be Cu-free by synchrotron X-ray photoelectron spectroscopy. The SLG grown by Cu-vapor-assisted CVD is fabricated into field effect transistor devices without transfer steps that are generally required when SLG is grown by regular CVD process on metal catalyst substrates. This method has overcome two important hurdles previously present when the catalyst-free CVD process is used for the growth of SLG on fused quartz and hexagonal boron nitride substrates, that is, high degree of structural defects and limited size of resulting graphene, respectively.